# A Maximum Likelihood Estimate of the mean for a multivariate normal distribution, the MLE of the mean is a scalar value or a vector scalar values?

In the following equation, I have found the maximum likelihood estimate of the mean for a multivariate normal distribution

$\therefore \mu^*_{MLE}=\dfrac{1}{n}\sum_{i=1}^{n}x_i$

But I am wondering if the $\mu$ is a scalar value or a vector scalar values.

• Is this the right way to think of what it is? $\mu_{MLE} = \begin{bmatrix}\mu_1\\\mu_2\\\vdots\\\mu_n\end{bmatrix} = \begin{bmatrix}\dfrac{1}{n}\sum_{i=1}^n x_1\\\dfrac{1}{n}\sum_{i=1}^n x_2\\\vdots\\\dfrac{1}{n}\sum_{i=1}^n x_i\end{bmatrix} = \begin{bmatrix}E(X_1)\\E(X_2)\\\vdots\\E(X_n)\end{bmatrix} = \begin{bmatrix}\bar{X_1} \\ \bar{X_2} \\ \vdots \\ \bar{X_n}\end{bmatrix}$ – user122358 Mar 12 '17 at 13:43

If $x_{i}$ is vector-valued, $\mu$ is a vector (one mean for each entry of $x_{i}$). You may consider the scalar case as a special case in which $\mathrm{dim}(x_{i})=1$.